White, correlated and fractal noise

Correlated random noise V(r) can be obtained by smoothing of the white noise f(r)
    V(r) = K(r-r') f(r') dr' ,
where f(r), V(r) are random function with correlators
    <f(r) f(r')> = δ(r-r') ,
    <V(r)V(r')> = K(r-s)K(r'-s) ds
K In the applet below you see 1D realisation of white and correlated noise with equidistant step in x. Independent random points fi with uniform distribution on interval (-1, 1) make the white noise (the blue curve). Correlated random points Vi (the red curve) are obtained by averaging of white noise in radius Rc sphere, i.e. kernel Ko is used
    Vi = ∑j= -Rc,Rc   fi+j

White and correlated random noises.
This noise for Rc = 32 and fractal one for m = 1.5 (see the third applet) are very similar. To get smooth random noise we shall take smooth kernel K(r). Gauss function
    KG(x) = exp[-(x/Rc)2]
is used below

Smooth correlated random noises.
Fractal noise. Press Enter to set a new m value.

To get a 2D fractal noise (mountain) you take an elastic string (see Fig.1), then a random vertical displacement is applied to its middle point. The process is repeated recursively to the middle point of every new segment. The random displacement decreases m times each iteration (usually m = 2 are used).
Fourier Using Fourier transformation for V(r), K(r), f(r)
    g(k) = g(r) eikr dr ,
we get for V(k)
    V(k) = K(k) f(k) .
I.e. averaging (*) means the white noise filtration by means of a filter with bandwidth K(k). The bandwidths for the two used filters are shown in Fig.3 (for Rc = 1)
    KG(k) ~ exp[-(Rck)2],     Ko ~ sin(Rck)/k.
Noise Vi for the kernel Ko is very broken because this transformation has broad bandwidth, decreasing only as 1/k for large k. We can use filtration (*) to the fractal noise too, to get more smooth "landscape".

At last 2D correlated random landscape. To get a smooth potential 2D Gauss kernel is used

Percolation in random potential landscape
Drag mouse to rotate 3D image (with "shift" to zoom it). The white line (in the blue bar to the right) corresponds to the average <V> value. The yellow line corresponds to the Fermi energy εF . Drag the line by mouse to change εF . See also 3D Mountains and Hidden Surface Removal Algorithms.

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updated 10 March 2002